Calculate symplectic bases for matrices over fields and the integers.¶
This module finds a symplectic basis for an anti-symmetric, alternating matrix M defined over a field or the integers.
Anti-symmetric means that \(M = -M^t\), where \(M^t\) denotes the transpose of \(M\). Alternating means that the diagonal of \(M\) is identically zero.
A symplectic basis is a basis of the form \(e_1, \ldots, e_j, f_1, \ldots, f_j, z_1, \ldots, z_k\) such that
- \(z_i M v^t\) = 0 for all vectors \(v\);
- \(e_i M {e_j}^t = 0\) for all \(i, j\);
- \(f_i M {f_j}^t = 0\) for all \(i, j\);
- \(e_i M {f_j}^t = 0\) for all \(i\) not equal \(j\);
and such that the non-zero terms
- \(e_i M {f_i}^t\) are “as nice as possible”: 1 over fields, or integers satisfying divisibility properties otherwise.
REFERENCES:
Bourbaki gives a nice proof that can be made constructive but is not efficient (see Section 5, Number 1, Theorem 1, page 79):
Bourbaki, N. Elements of Mathematics, Algebra III, Springer Verlag 2007.
Kuperburg gives a more efficient and constructive exposition (see Theorem 18).
Kuperberg, Greg. Kasteleyn Cokernels. Electr. J. Comb. 9(1), 2002.
TODO:
The routine over the integers applies over general principal ideal domains.
WARNING:
This code is not a good candidate for conversion to Cython. The
majority of the execution time is spent adding multiples of
columns and rows, which is already fast. It would be better to
devise a better algorithm, perhaps modular or based on a fast
smith_form implementation.
AUTHOR:
- Nick Alexander: initial implementation
- David Loeffler (2008-12-08): changed conventions for consistency with smith_form
-
sage.matrix.symplectic_basis.symplectic_basis_over_ZZ(M)¶ Find a symplectic basis for an anti-symmetric, alternating matrix M defined over the integers.
Returns a pair (F, C) such that the rows of C form a symplectic basis for M and F = C * M * C.transpose().
Anti-symmetric means that \(M = -M^t\). Alternating means that the diagonal of \(M\) is identically zero.
A symplectic basis is a basis of the form \(e_1, \ldots, e_j, f_1, \ldots, f_j, z_1, \ldots, z_k\) such that
- \(z_i M v^t\) = 0 for all vectors \(v\);
- \(e_i M {e_j}^t = 0\) for all \(i, j\);
- \(f_i M {f_j}^t = 0\) for all \(i, j\);
- \(e_i M {f_i}^t = d_i\) for all \(i\), where d_i are positive integers such that \(d_{i} | d_{i+1}\) for all \(i\);
- \(e_i M {f_j}^t = 0\) for all \(i\) not equal \(j\).
The ordering for the factors \(d_{i} | d_{i+1}\) and for the placement of zeroes was chosen to agree with the output of
smith_form.See the examples for a pictorial description of such a basis.
EXAMPLES:
sage: from sage.matrix.symplectic_basis import symplectic_basis_over_ZZ
An example which does not have full rank:
sage: E = matrix(ZZ, 4, 4, [0, 16, 0, 2, -16, 0, 0, -4, 0, 0, 0, 0, -2, 4, 0, 0]); E [ 0 16 0 2] [-16 0 0 -4] [ 0 0 0 0] [ -2 4 0 0] sage: F, C = symplectic_basis_over_ZZ(E) sage: F [ 0 2 0 0] [-2 0 0 0] [ 0 0 0 0] [ 0 0 0 0] sage: C * E * C.transpose() == F True
A larger example:
sage: E = matrix(ZZ, 8, 8, [0, 25, 0, 0, -37, -3, 2, -5, -25, 0, 1, -5, -54, -3, 3, 3, 0, -1, 0, 7, 0, -4, -20, 0, 0, 5, -7, 0, 0, 14, 0, -3, 37, 54, 0, 0, 0, 2, 3, -12, 3, 3, 4, -14, -2, 0, -3, 2, -2, -3, 20, 0, -3, 3, 0, -2, 5, -3, 0, 3, 12, -2, 2, 0]); E [ 0 25 0 0 -37 -3 2 -5] [-25 0 1 -5 -54 -3 3 3] [ 0 -1 0 7 0 -4 -20 0] [ 0 5 -7 0 0 14 0 -3] [ 37 54 0 0 0 2 3 -12] [ 3 3 4 -14 -2 0 -3 2] [ -2 -3 20 0 -3 3 0 -2] [ 5 -3 0 3 12 -2 2 0] sage: F, C = symplectic_basis_over_ZZ(E) sage: F [ 0 0 0 0 1 0 0 0] [ 0 0 0 0 0 1 0 0] [ 0 0 0 0 0 0 1 0] [ 0 0 0 0 0 0 0 20191] [ -1 0 0 0 0 0 0 0] [ 0 -1 0 0 0 0 0 0] [ 0 0 -1 0 0 0 0 0] [ 0 0 0 -20191 0 0 0 0] sage: F == C * E * C.transpose() True sage: E.smith_form()[0] [ 1 0 0 0 0 0 0 0] [ 0 1 0 0 0 0 0 0] [ 0 0 1 0 0 0 0 0] [ 0 0 0 1 0 0 0 0] [ 0 0 0 0 1 0 0 0] [ 0 0 0 0 0 1 0 0] [ 0 0 0 0 0 0 20191 0] [ 0 0 0 0 0 0 0 20191]
An odd dimensional example:
sage: E = matrix(ZZ, 5, 5, [0, 14, 0, -8, -2, -14, 0, -3, -11, 4, 0, 3, 0, 0, 0, 8, 11, 0, 0, 8, 2, -4, 0, -8, 0]); E [ 0 14 0 -8 -2] [-14 0 -3 -11 4] [ 0 3 0 0 0] [ 8 11 0 0 8] [ 2 -4 0 -8 0] sage: F, C = symplectic_basis_over_ZZ(E) sage: F [ 0 0 1 0 0] [ 0 0 0 2 0] [-1 0 0 0 0] [ 0 -2 0 0 0] [ 0 0 0 0 0] sage: F == C * E * C.transpose() True sage: E.smith_form()[0] [1 0 0 0 0] [0 1 0 0 0] [0 0 2 0 0] [0 0 0 2 0] [0 0 0 0 0] sage: F.parent() Full MatrixSpace of 5 by 5 dense matrices over Integer Ring sage: C.parent() Full MatrixSpace of 5 by 5 dense matrices over Integer Ring
-
sage.matrix.symplectic_basis.symplectic_basis_over_field(M)¶ Find a symplectic basis for an anti-symmetric, alternating matrix M defined over a field.
Returns a pair (F, C) such that the rows of C form a symplectic basis for M and
F = C * M * C.transpose().Anti-symmetric means that \(M = -M^t\). Alternating means that the diagonal of \(M\) is identically zero.
A symplectic basis is a basis of the form \(e_1, \ldots, e_j, f_1, \ldots f_j, z_1, \ldots, z_k\) such that
- \(z_i M v^t\) = 0 for all vectors \(v\);
- \(e_i M {e_j}^t = 0\) for all \(i, j\);
- \(f_i M {f_j}^t = 0\) for all \(i, j\);
- \(e_i M {f_i}^t = 1\) for all \(i\);
- \(e_i M {f_j}^t = 0\) for all \(i\) not equal \(j\).
See the examples for a pictorial description of such a basis.
EXAMPLES:
sage: from sage.matrix.symplectic_basis import symplectic_basis_over_field
A full rank exact example:
sage: E = matrix(QQ, 8, 8, [0, -1/2, -2, 1/2, 2, 0, -2, 1, 1/2, 0, -1, -3, 0, 2, 5/2, -3, 2, 1, 0, 3/2, -1, 0, -1, -2, -1/2, 3, -3/2, 0, 1, 3/2, -1/2, -1/2, -2, 0, 1, -1, 0, 0, 1, -1, 0, -2, 0, -3/2, 0, 0, 1/2, -2, 2, -5/2, 1, 1/2, -1, -1/2, 0, -1, -1, 3, 2, 1/2, 1, 2, 1, 0]); E [ 0 -1/2 -2 1/2 2 0 -2 1] [ 1/2 0 -1 -3 0 2 5/2 -3] [ 2 1 0 3/2 -1 0 -1 -2] [-1/2 3 -3/2 0 1 3/2 -1/2 -1/2] [ -2 0 1 -1 0 0 1 -1] [ 0 -2 0 -3/2 0 0 1/2 -2] [ 2 -5/2 1 1/2 -1 -1/2 0 -1] [ -1 3 2 1/2 1 2 1 0] sage: F, C = symplectic_basis_over_field(E); F [ 0 0 0 0 1 0 0 0] [ 0 0 0 0 0 1 0 0] [ 0 0 0 0 0 0 1 0] [ 0 0 0 0 0 0 0 1] [-1 0 0 0 0 0 0 0] [ 0 -1 0 0 0 0 0 0] [ 0 0 -1 0 0 0 0 0] [ 0 0 0 -1 0 0 0 0] sage: F == C * E * C.transpose() True
An example over a finite field:
sage: E = matrix(GF(7), 8, 8, [0, -1/2, -2, 1/2, 2, 0, -2, 1, 1/2, 0, -1, -3, 0, 2, 5/2, -3, 2, 1, 0, 3/2, -1, 0, -1, -2, -1/2, 3, -3/2, 0, 1, 3/2, -1/2, -1/2, -2, 0, 1, -1, 0, 0, 1, -1, 0, -2, 0, -3/2, 0, 0, 1/2, -2, 2, -5/2, 1, 1/2, -1, -1/2, 0, -1, -1, 3, 2, 1/2, 1, 2, 1, 0]); E [0 3 5 4 2 0 5 1] [4 0 6 4 0 2 6 4] [2 1 0 5 6 0 6 5] [3 3 2 0 1 5 3 3] [5 0 1 6 0 0 1 6] [0 5 0 2 0 0 4 5] [2 1 1 4 6 3 0 6] [6 3 2 4 1 2 1 0] sage: F, C = symplectic_basis_over_field(E); F [0 0 0 0 1 0 0 0] [0 0 0 0 0 1 0 0] [0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 1] [6 0 0 0 0 0 0 0] [0 6 0 0 0 0 0 0] [0 0 6 0 0 0 0 0] [0 0 0 6 0 0 0 0] sage: F == C * E * C.transpose() True
The tricky case of characteristic 2:
sage: E = matrix(GF(2), 8, 8, [0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0]); E [0 0 1 1 0 1 0 1] [0 0 0 0 0 0 0 0] [1 0 0 0 0 0 1 1] [1 0 0 0 0 0 0 1] [0 0 0 0 0 1 1 0] [1 0 0 0 1 0 1 1] [0 0 1 0 1 1 0 0] [1 0 1 1 0 1 0 0] sage: F, C = symplectic_basis_over_field(E); F [0 0 0 1 0 0 0 0] [0 0 0 0 1 0 0 0] [0 0 0 0 0 1 0 0] [1 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0] [0 0 1 0 0 0 0 0] [0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0] sage: F == C * E * C.transpose() True
An inexact example:
sage: E = matrix(RR, 8, 8, [0.000000000000000, 0.420674846479344, -0.839702420666807, 0.658715385244413, 1.69467394825853, -1.14718543053828, 1.03076138152950, -0.227739521708484, -0.420674846479344, 0.000000000000000, 0.514381455379082, 0.282194064028260, -1.38977093018412, 0.278305070958958, -0.0781320488361574, -0.496003664217833, 0.839702420666807, -0.514381455379082, 0.000000000000000, -0.00618222322875384, -0.318386939149028, -0.0840205427053993, 1.28202592892333, -0.512563654267693, -0.658715385244413, -0.282194064028260, 0.00618222322875384, 0.000000000000000, 0.852525732369211, -0.356957405431611, -0.699960114607661, 0.0260496330859998, -1.69467394825853, 1.38977093018412, 0.318386939149028, -0.852525732369211, 0.000000000000000, -0.836072521423577, 0.450137632758469, -0.696145287292091, 1.14718543053828, -0.278305070958958, 0.0840205427053993, 0.356957405431611, 0.836072521423577, 0.000000000000000, 0.214878541347751, -1.20221688928379, -1.03076138152950, 0.0781320488361574, -1.28202592892333, 0.699960114607661, -0.450137632758469, -0.214878541347751, 0.000000000000000, 0.785074452163036, 0.227739521708484, 0.496003664217833, 0.512563654267693, -0.0260496330859998, 0.696145287292091, 1.20221688928379, -0.785074452163036, 0.000000000000000]); E [ 0.000000000000000 0.420674846479344 -0.839702420666807 0.658715385244413 1.69467394825853 -1.14718543053828 1.03076138152950 -0.227739521708484] [ -0.420674846479344 0.000000000000000 0.514381455379082 0.282194064028260 -1.38977093018412 0.278305070958958 -0.0781320488361574 -0.496003664217833] [ 0.839702420666807 -0.514381455379082 0.000000000000000 -0.00618222322875384 -0.318386939149028 -0.0840205427053993 1.28202592892333 -0.512563654267693] [ -0.658715385244413 -0.282194064028260 0.00618222322875384 0.000000000000000 0.852525732369211 -0.356957405431611 -0.699960114607661 0.0260496330859998] [ -1.69467394825853 1.38977093018412 0.318386939149028 -0.852525732369211 0.000000000000000 -0.836072521423577 0.450137632758469 -0.696145287292091] [ 1.14718543053828 -0.278305070958958 0.0840205427053993 0.356957405431611 0.836072521423577 0.000000000000000 0.214878541347751 -1.20221688928379] [ -1.03076138152950 0.0781320488361574 -1.28202592892333 0.699960114607661 -0.450137632758469 -0.214878541347751 0.000000000000000 0.785074452163036] [ 0.227739521708484 0.496003664217833 0.512563654267693 -0.0260496330859998 0.696145287292091 1.20221688928379 -0.785074452163036 0.000000000000000] sage: F, C = symplectic_basis_over_field(E); F # random [ 0.000000000000000 0.000000000000000 2.22044604925031e-16 -2.22044604925031e-16 1.00000000000000 0.000000000000000 0.000000000000000 -3.33066907387547e-16] [ 0.000000000000000 8.14814392305203e-17 -1.66533453693773e-16 -1.11022302462516e-16 0.000000000000000 1.00000000000000 -1.11022302462516e-16 0.000000000000000] [-5.27829526256056e-16 -2.40004077757759e-16 1.28373418199470e-16 -1.11022302462516e-16 0.000000000000000 -3.15483812822081e-16 1.00000000000000 -4.44089209850063e-16] [ 1.31957381564014e-16 1.41622049084608e-16 -6.68515202578511e-17 -3.95597468756028e-17 -4.85722573273506e-17 -5.32388011580111e-17 -1.31328455615552e-16 1.00000000000000] [ -1.00000000000000 0.000000000000000 0.000000000000000 4.85722573273506e-17 0.000000000000000 -5.55111512312578e-17 -1.11022302462516e-16 2.22044604925031e-16] [ 0.000000000000000 -1.00000000000000 0.000000000000000 -2.77555756156289e-17 5.55111512312578e-17 -8.69223574327834e-17 0.000000000000000 -4.44089209850063e-16] [ 0.000000000000000 -1.05042437087238e-17 -1.00000000000000 3.33066907387547e-16 1.11022302462516e-16 -1.18333563634309e-16 4.40064433050777e-17 2.22044604925031e-16] [ 5.27829526256056e-16 1.99901485752317e-16 1.65710718121313e-17 -1.00000000000000 -2.22044604925031e-16 5.52150940090699e-16 -3.93560383111738e-16 1.01155762925061e-16] sage: F == C * E * C.transpose() True sage: abs(F[0, 4] - 1) < 1e-10 True sage: abs(F[4, 0] + 1) < 1e-10 True sage: F.parent() Full MatrixSpace of 8 by 8 dense matrices over Real Field with 53 bits of precision sage: C.parent() Full MatrixSpace of 8 by 8 dense matrices over Real Field with 53 bits of precision